在时间分布漂移下，用小波阈值法实现最优估计，无需先验知识。
Adaptive Estimation and Learning under Temporal Distribution Shift
- 基于小波变换的软阈值法，自动适应时间序列非平稳程度。
- 在无先验信息下达到最优点估计误差率，理论与实验均验证。
- 适用于分类风险分析和高效训练目标设计，对信号处理有启发。
本文研究时间分布漂移下的估计与学习问题。考虑一个长度为 $n$ 的观测序列，它是随时间变化的真实序列的噪声实现。目标是估计最终时刻的真实值，并提供精确的逐点估计误差率。我们证明，在缺乏时间漂移程度先验知识的情况下，小波软阈值估计器可达到真实值的最优估计误差界。所提方法通过建立序列非平稳性水平与小波域稀疏性之间的联系，推广了现有研究（Mazzetto and Upfal, 2023）。理论结果经数值实验验证。进一步将该估计器应用于分布漂移下的二分类任务，推导出稀疏感知的超额风险界，并设计了计算高效的训练目标。最后，揭示了本结果与经典变差去噪问题（Mammen and van de Geer, 1997; Tibshirani, 2014）的关联，提出了此类任务的新最优算法。
原文摘要 · Abstract (English)
In this paper, we study the problem of estimation and learning under temporal distribution shift. Consider an observation sequence of length $n$, which is a noisy realization of a time-varying groundtruth sequence. Our focus is to develop methods to estimate the groundtruth at the final time-step while providing sharp point-wise estimation error rates. We show that, without prior knowledge on the level of temporal shift, a wavelet soft-thresholding estimator provides an optimal estimation error bound for the groundtruth. Our proposed estimation method generalizes existing researches Mazzetto and Upfal (2023) by establishing a connection between the sequence's non-stationarity level and the sparsity in the wavelet-transformed domain. Our theoretical findings are validated by numerical experiments. Additionally, we applied the estimator to derive sparsity-aware excess risk bounds for binary classification under distribution shift and to develop computationally efficient training objectives. As a final contribution, we draw parallels between our results and the classical signal processing problem of total-variation denoising (Mammen and van de Geer,1997; Tibshirani, 2014), uncovering novel optimal algorithms for such task.
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